Completeness of exponentials and Beurling's theorem regarding Fourier transform on Rn\mathbb{R}^n and Tn\mathbb{T}^n

A classical result of A. Beurling gives a relation between the decay of a complex Borel measure on $\mathbb{R}$ and the vanishing set of its Fourier transform. We prove several variable analogues of this result on the Euclidean space $\mathbb{R}^n$ and the $n$-dimensional torus $\mathbb{T}^n$. We also prove some results on the well known weighted approximation problem of exponentials on $\mathbb{R}^n$ and $\mathbb{T}^n$ by establishing an equivalence with Beurling's theorem.Comment: Lemma 3.2 has been replaced with an updated version and the proof of Theorem 3.4 has been changed accordingl